[seqfan] D(n)

[David Sycamore]

Hello Seqfans,
For n =d_1 d_2...d_k, define D(n) to be the unique number formed by adding all identical digits of n in order of appearance in n. (See draft A330273 for details). Example D(224)=44 and D(44)=8.

Let e(n) be the smallest number such that D*e(n)(n) can be reduced no further (then e(224)=2). If the digits of n are all distinct, then e(n)=0, since D(n)=n.

Let a(n) be the smallest number k such that e(k)=n; (n>=0). Does a(n) always exist?

I am trying to find some terms. So far I have candidates for a(n) up to n=7. (by hand) :

1, 11, 112, 1124, 11248, 122468, 2486, 122468..... ?

But some of these may be wrong, I’m not sure....

If anyone feels like computing the right numbers please have a go
Thanks,
David.